Suppose that we are studying a system that we know conserves energy.
In applications of DMD, it is implicitly assumed that measurements of the
state have been suitably weighted so that the
square of the 2-norm corresponds to the energy of the state [1]}: \(E(x) = \Vert  x\Vert _2^2\).
In these variables,
the original optimization problems (REF , REF )
equate to finding the model \(A\)  that minimises the energy of the error between
the true and predicted states (\(y_k\)  and \(Ax_k\)  respectively).
Thus, if \(A\)  represents a discrete-time linear dynamical system
(\(y_k = x_{k+1}= Ax_k\) )
then \(A\)  is an energy preserving operator if and only if
\(E(Ax)= \Vert  Ax\Vert _2^2 = \Vert x\Vert _2^2 = E(x) \qquad \textnormal {for all } x\in \mathbb {R}^n.\) 
